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posts/clt-intuitive-derivation/index.qmd

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The **Central Limit Theorem (CLT)** answers an important question: Why does the **bell curve** (or Normal Distribution) show up everywhere in the real world?
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![The Normal Distribution, showing the 68-95-99.7 rule.](images/normal_distribution.png){fig-alt="Normal Distribution bell curve diagram showing the 68-95-99.7 rule with standard deviations from the mean" width="600"}
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![The Normal Distribution with it's pdf.](images/normal_distribution.png){fig-alt="Normal Distribution bell curve diagram" width="600"}
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Specifically, the theorem describes what happens when you take a random variable, $X$, and repeat the experiment many times to get a series of outcomes, $X_1, X_2, \dots, X_m$. What does the distribution of their **sum** ($S_m = X_1 + \dots + X_m$) or their **average** ($\bar{X}_m = S_m/m$) look like when $m$ is very large?
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